On the p-norm of the truncated n-dimensional Hilbert transform

Abstract

It is shown that a bounded linear operator T from L�(Rn) to itself which commutes both with translations and dilatations is a finite linear combination of Hilbert-type transforms. Using this we show that the ρ-norm of the Hilbert transform is the same as the ρ-norm of its truncation to any Lebesgue measurable subset of Rn with non-zero measure.